Can you add a little v symbol option for veterans and have it under their name or beside it
As Family Tree has international participation I don't see this as a particularly good idea. Each country has the right to be proud of its veterans, but there would not necessarily be the same kind of appreciation from the citizens in which country the veteran might have served.
This comment might appear to be political, but I believe highlighting "veteran" status would be a divisive act, when considering our international community.
I’m meaning the person that’s controlling the deceased person can add or remove the symbol
The FamilySearch Family Tree is fully open-edit. That means anyone can edit or change the profile. There is no "controlling the deceased person" in the FSFT.
I meant that I wasn’t trying to go into fine details
I would agree with the comment @Paul W added.
FamilySearch has global participation. The view of a person who has served in a nation's military may not viewed the same in all countries. And indeed could have political implications in some.
Additionally, the American English use of the word "veteran" is somewhat different to how it is used in other parts of the English speaking world, where it can mean quite different things. In particular, it is perhaps more commonly interpreted as an older person, similar to what in American English may be described as a "Senior".
It would surely require some sort of source tag to make it verifiable, and if there are relevant sources in place, e.g. military pension information, then surely the person's status in this regard is clear anyhow to anyone viewing the profile?
One of my pet peeves involves the recent addition of that same V to memorials on findagrave.com. Some contributors have decided it is their mission to ask for changes to all the memorials that are not so marked. Of course, it was not possible to add that V when the memorials were created, and many of them are in Veterans-only cemeteries. I'm fielding dozens of requests to add the V daily.